发表机构
Shanghai Jiao Tong University（上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了非整数阶Rényi熵估计的样本复杂度，给出了充分且必要的样本数上界和下界，完成了Rényi熵估计复杂度图景。
AI 中文摘要
Rényi熵估计已被Acharya、Orlitsky、Suresh和Tyagi（SODA 2015；IEEE Trans. Inf. Theory 2017）及其后续工作全面研究，然而只有整数阶Rényi熵估计的样本复杂度已被解决。在本文中，我们解决了非整数阶Rényi熵估计的样本复杂度，从而完成了Rényi熵估计的复杂度图景。具体而言，我们证明对于任何非整数$\alpha > 0$，使用 \\[ \Theta\\!\left(\frac{d^{\max\{1/\alpha,1\}}}{\varepsilon^{1/\alpha}\log(d)} + \frac{d^{|1-1/\alpha|}}{\varepsilon^2}\right) \\] 个样本，足以且必要地估计字母表大小为$d$的未知离散分布的$\alpha$阶Rényi熵，误差在加法误差$\varepsilon$以内。对于上界，我们使用一种精细的多项式近似估计器来减少大概率的偏差。对于下界，我们采用一个不同的困难实例，并配备一种新的矩匹配构造。该构造性矩匹配具有常数有界的高阶矩，同时达到$\alpha$阶矩之间的固定比率，这具有独立的意义。
英文摘要
Rényi entropy estimation has been comprehensively investigated by Acharya, Orlitsky, Suresh and Tyagi (SODA 2015; IEEE Trans. Inf. Theory 2017) and consequent works, whereas only the sample complexity of Rényi entropy estimation of integer order has been settled. In this paper, we settle the sample complexity of Rényi entropy estimation of noninteger order, thereby completing the complexity picture of Rényi entropy estimation. Specifically, we show that for any noninteger $α> 0$, it is sufficient and necessary to use \[ Θ\!\left(\frac{d^{\max\{1/α,1\}}}{\varepsilon^{1/α}\log(d)} + \frac{d^{|1-1/α|}}{\varepsilon^2}\right) \] samples to estimate the Rényi entropy of order $α$ of an unknown discrete distribution over an alphabet of size $d$ to within additive error $\varepsilon$. For the upper bound, we reduce the bias using a refined polynomial approximation estimator for large probabilities. For the lower bound, we employ a different hard instance equipped with a new moment matching construction. The constructive moment matching has constant bounded high-order moments, while attaining a fixed ratio between the $α$-th moments, which is of independent interest.
Comments23 pages, 1 table
